曾空 取半径厂处宽度为d的微小环形面积,则可得流量为 R Q=|,d= (R-r purr=mpay Apa (55) A 04 8l1281 此式称为哈根一伯泊肃叶( Hagen- Poiseuille)定律。 表明: 层流时管中流量与管半径或直径的四次方成比例。 26
26 取半径 r 处宽度为 dr 的微小环形面积，则可得流量为 ( ) l pd l pd R r rdr l p Q dA A R y 8 128 2 4 4 4 2 0 = − = = = （5—5） 此式称为哈根—伯泊肃叶 (agen－Ρoisuille) 定律。 表明： 层流时管中流量与管半径或直径的四次方成比例
Flowing in pipe 3. Average velocity and the greatest velocity average velocity in pipe 4 Q PApR R (56) a burR 8ul The most velocity in pipe is on the axes where r=0. Get from the formula (5-4) y=0 max ApR_-=2t aLl On engineering applying this speciality of laminar flow to measure the velocity of flow at axes and calculate the flux directly is very convenient 27
27 3. Average velocity and the greatest velocity 2 2 4 8 8 R l p l R pR A Q = = = （5—6） The most velocity in pipe is on the axes where r = 0. Get from the formula (5—4） r = 0 2 4 2 max = = l pR On engineering applying this speciality of laminar flow to measure the velocity of flow at axes and calculate the flux directly is very convenient . average velocity in pipe:
着空 三、平均速度和最大速度 管中平均速度 2- DApR_Ap R (56) a buRr 8ul 管中最大速度在轴心F=0处,由式(54)得 max ApR_-=2t aLl 工程上应用层流这一特性直接从测定管轴心处流速而计算流 量相当方便。 28
28 三、平均速度和最大速度 2 2 4 8 8 R l p l R pR A Q = = = （5—6） 管中最大速度在轴心 r = 0 处，由式（5—4）得 2 4 2 max = = l pR 工程上应用层流这一特性直接从测定管轴心处流速而计算流 量相当方便。 管中平均速度
Flowing in pipe 4. Distribution of shear force According to the Newtons internal friction law, we can get in round pipe du dU,△Pr z=±y=H (5—7) db dr 2l Obtain from the formula (5-7) T=Tr (5-8) △DR when r=ro to the shear force on the pipe well is △DR (59) then explain: R On the cross section of laminar flow the shear force is in direct ratio with the radius. The distribution rule is shown in Figure 5-6 which is called k font distribution. 29
29 4. Distribution of shear force dr l d dy d y y 2 Pr = = − = （5—7） Obtain from the formula（5—7） =(r) （5—8） when ，the shear force on the pipe well is l pR r r 2 0 0 = = According to the Newton’s internal friction law , we can get in round pipe explain： On the cross section of laminar flow the shear force is in direct ratio with the radius. The distribution rule is shown in Figure 5—6， which is called k font distribution. l pR 2 0 = （5—9） then R r = 0
着空 四、切应力分布 根据牛顿内摩擦定律,在圆管中可得 au △Pr =+,=-1 (5—7) dr 2/ 由式(5-7)可得 z=() (5-8) 当r=60o=时,可得管壁处的切应力为 21 ◆pR 021 (59) 则 0 R 说明: 在层流的过流断面上,切应力与半径成正比。分布规律如图 56所示,称为切应力的k字形分布
30 四、切应力分布 dr l d dy d y y 2 Pr = = − = （5—7） 由式（5—7）可得 =(r) （5—8） 当 时，可得管壁处的切应力为 l pR r r 2 0 0 = = 根据牛顿内摩擦定律，在圆管中可得 说明： 在层流的过流断面上，切应力与半径成正比。分布规律如图 5—6所示，称为切应力的 k 字形分布。 l pR 2 0 = （5—9） 则 R r = 0